A Lagrangian approach for the compressible Navier-Stokes equations

Existence of strong solutions with critical regularity to a polytropic model for radiating flows

“…They are located either on the diagonal (and are positive if ν > γ > 0 and κ > 0) or in the blocks B 12 2 and B 21 2 that correspond to interactions between the (modified) fluid unknowns t V 1 := ( a, d, ) and radiative unknowns t V 2 := ( j 0 , j 1 ). Therefore an important part of the stability analysis will be dedicated to the 3×3 subsystem with matrix A 11 1 ρ + B 11 2 ρ 2 satisfied by V 1 , and to the 2 × 2 subsystem with matrix + ρ A 22 1 + ρ 2 B 22 2 fulfilled by V 2 . For both sub-systems, interactions between the fluid unknowns V 1 and radiative unknowns V 2 will be considered as error terms in the right-hand side, that may be eliminated for small enough ρ.…”

“…We shall in particular exhibit a necessary and sufficient linear stability condition in the low-frequency regime (which is fulfilled in the strongly relativistic regime), and prove optimal global-in-time estimates for the linearized equations. The next section is devoted to the proof of similar estimates for the so-called paralinearized system (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15). Those estimates are the key to our global existence result and to the rigorous justification of the nonrelativistic limit (see Sect.…”

“…(15) The boundary conditions at infinity (e.g., convergence to some stable constant state) will be implicitly given by the functional framework we shall work in. System (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) has been investigated recently in astrophysics and laser applications (in the relativistic and inviscid case) by Lowrie, Morel and Hittinger [24], Buet and Després [3], with a special attention to asymptotic regimes. The global existence result of weak solutions has been established by Ducomet et al [15].…”

“…In the next section, we write out the system for the P1 approximation of System (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) in the grey case, and state our main results: first local-in-time well posedness either for smooth data and quite general assumptions on the coefficients of the system, or in the "critical regularity framework" but for coefficients depending only on the density, and linear equilibrium distribution function; second, global existence for small perturbations of a stable constant equilibrium in the strongly underrelativistic situation. Section 3 is devoted to the spectral analysis of the linearized equations about a constant reference state.…”

Fourier Analysis Methods for the Compressible Navier-Stokes Equations

“…It is now easy to conclude to Theorem 3.1 in its full generality, as a mere corollary of the following proposition which states the equivalence of Systems (26) and (50) in our functional setting (see the proof in [16]). …”